Precalculus Trigonometry Review – Step-by-Step Guidance

Q1. Simplify: \( \frac{1 + \cos(2\omega)}{\cos \omega} \)

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to use double-angle identities and algebraic manipulation to simplify trigonometric expressions.

Key Terms and Formulas

• Double-angle identity for cosine: $\cos(2\theta) = 2\cos^2\theta - 1$ or $\cos(2\theta) = 1 - 2\sin^2\theta$

• Basic algebraic manipulation

Step-by-Step Guidance

1. Recall the double-angle identity for cosine: $\cos(2\omega) = 2\cos^2\omega - 1$.

2. Substitute this identity into the numerator: $1 + (2\cos^2\omega - 1)$.

3. Simplify the numerator by combining like terms.

4. Write the new numerator over $\cos \omega$ and see if further simplification is possible (such as canceling terms).

Try solving on your own before revealing the answer!

Q2. Simplify: \( \frac{\csc^2 x - 1}{\cot x} \)

Background

Topic: Trigonometric Identities and Simplification

This question tests your knowledge of Pythagorean identities and your ability to manipulate trigonometric expressions.

Key Terms and Formulas

• Pythagorean identity: $\csc^2 x = 1 + \cot^2 x$

• Quotient identities: $\cot x = \frac{\cos x}{\sin x}$, $\csc x = \frac{1}{\sin x}$

Step-by-Step Guidance

1. Substitute the Pythagorean identity: $\csc^2 x - 1 = \cot^2 x$.

2. Rewrite the numerator as $\cot^2 x$ over $\cot x$.

3. Simplify the fraction $\frac{\cot^2 x}{\cot x}$.

4. Express your answer in terms of basic trigonometric functions if possible.

Try solving on your own before revealing the answer!

Q3. Simplify: \( \frac{\sin^2 x + \cos^2 x}{1 + \cot^2 x} \)

Background

Topic: Pythagorean Identities and Simplification

This question checks your understanding of fundamental trigonometric identities and algebraic simplification.

Key Terms and Formulas

• Pythagorean identity: $\sin^2 x + \cos^2 x = 1$

• Another Pythagorean identity: $1 + \cot^2 x = \csc^2 x$

Step-by-Step Guidance

1. Apply the Pythagorean identity to the numerator: $\sin^2 x + \cos^2 x = 1$.

2. Apply the Pythagorean identity to the denominator: $1 + \cot^2 x = \csc^2 x$.

3. Rewrite the expression as $\frac{1}{\csc^2 x}$.

4. Recall that $\csc x = \frac{1}{\sin x}$, so $\csc^2 x = \frac{1}{\sin^2 x}$, and simplify further.

Try solving on your own before revealing the answer!

Q4. Simplify: \( \frac{\sec x}{\cos x \sin x} \)

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to rewrite trigonometric expressions using reciprocal and quotient identities.

Key Terms and Formulas

• Reciprocal identities: $\sec x = \frac{1}{\cos x}$

• Product of trigonometric functions

Step-by-Step Guidance

1. Rewrite $\sec x$ as $\frac{1}{\cos x}$.

2. Substitute into the expression: $\frac{1}{\cos x \cdot \cos x \sin x}$.

3. Simplify the denominator to $\cos^2 x \sin x$.

4. Express the entire fraction as a single term with powers of sine and cosine.

Try solving on your own before revealing the answer!

Q5. Evaluate: $\sin(45^\circ + 30^\circ)$

Background

Topic: Sum and Difference Formulas for Sine

This question tests your ability to use the sine addition formula and evaluate sine and cosine at special angles.

Key Terms and Formulas

• Sine addition formula: $\sin(A + B) = \sin A \cos B + \cos A \sin B$

• Values: $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$

Step-by-Step Guidance

1. Write the sum as $\sin(45^\circ + 30^\circ)$.

2. Apply the sine addition formula: $\sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$.

3. Substitute the known values for each trigonometric function.

4. Simplify the resulting expression by multiplying and adding the terms.

Try solving on your own before revealing the answer!

Q6. Evaluate: $\tan(60^\circ - 45^\circ)$

Background

Topic: Sum and Difference Formulas for Tangent

This question tests your ability to use the tangent difference formula and evaluate tangent at special angles.

Key Terms and Formulas

• Tangent difference formula: $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

• Values: $\tan 60^\circ = \sqrt{3}$, $\tan 45^\circ = 1$

Step-by-Step Guidance

1. Write the difference as $\tan(60^\circ - 45^\circ)$.

2. Apply the tangent difference formula: $\frac{\tan 60^\circ - \tan 45^\circ}{1 + \tan 60^\circ \tan 45^\circ}$.

3. Substitute the known values for $\tan 60^\circ$ and $\tan 45^\circ$.

4. Simplify the numerator and denominator separately before combining into a single fraction.

Try solving on your own before revealing the answer!

Q7. Simplify: $\cos(x + 180^\circ) + \cos x$

Background

Topic: Trigonometric Function Properties

This question tests your understanding of the periodicity and symmetry of the cosine function.

Key Terms and Formulas

• Cosine periodicity: $\cos(x + 180^\circ) = -\cos x$

Step-by-Step Guidance

1. Recall the property: $\cos(x + 180^\circ) = -\cos x$.

2. Substitute this into the expression: $-\cos x + \cos x$.

3. Simplify the sum.

Try solving on your own before revealing the answer!

Q8. Rewrite as a single trig function: $\sin x \cos 45^\circ + \cos x \sin 45^\circ$

Background

Topic: Sum and Difference Formulas for Sine

This question tests your ability to recognize and use the sine addition formula in reverse.

Key Terms and Formulas

• Sine addition formula: $\sin A \cos B + \cos A \sin B = \sin(A + B)$

Step-by-Step Guidance

1. Identify $A = x$ and $B = 45^\circ$ in the formula.

2. Recognize that the expression matches the sine addition formula.

3. Rewrite the expression as $\sin(x + 45^\circ)$.

Try solving on your own before revealing the answer!

Q9. Evaluate: $\sin(2 \cdot 75^\circ)$

Background

Topic: Double-Angle Formula for Sine

This question tests your ability to use the double-angle formula and evaluate sine and cosine at special angles.

Key Terms and Formulas

• Double-angle formula: $\sin(2\theta) = 2\sin \theta \cos \theta$

• Values: $\sin 75^\circ$, $\cos 75^\circ$ (may need to use sum formulas to find these)

Step-by-Step Guidance

1. Let $\theta = 75^\circ$, so $\sin(2\theta) = \sin(150^\circ)$.

2. Apply the double-angle formula: $2\sin 75^\circ \cos 75^\circ$.

3. Find $\sin 75^\circ$ and $\cos 75^\circ$ using sum formulas if needed.

4. Multiply the values and simplify.

Try solving on your own before revealing the answer!

Q10. Evaluate: $\cos(15^\circ)$

Background

Topic: Sum and Difference Formulas for Cosine

This question tests your ability to use the cosine difference formula and evaluate cosine and sine at special angles.

Key Terms and Formulas

• Cosine difference formula: $\cos(A - B) = \cos A \cos B + \sin A \sin B$

• Values: $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 30^\circ = \frac{1}{2}$

Step-by-Step Guidance

1. Express $15^\circ$ as $45^\circ - 30^\circ$.

2. Apply the cosine difference formula: $\cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ$.

3. Substitute the known values for each function.

4. Simplify the resulting expression.

Try solving on your own before revealing the answer!

Q11. Solve on $[0, 2\pi)$: $2 \sin x = \sqrt{3}$

Background

Topic: Solving Basic Trigonometric Equations

This question tests your ability to solve for $x$ in a basic sine equation within a given interval.

Key Terms and Formulas

• Inverse sine: $x = \arcsin(y)$

• Unit circle values for sine

Step-by-Step Guidance

1. Divide both sides by 2 to isolate $\sin x$.

2. Find all $x$ in $[0, 2\pi)$ such that $\sin x$ equals the resulting value.

3. Recall that sine is positive in quadrants I and II, so find both solutions in the interval.

Try solving on your own before revealing the answer!

Q12. Solve on $[0, 2\pi)$: $2 \sin^2 x - \sin x - 1 = 0$

Background

Topic: Solving Quadratic Trigonometric Equations

This question tests your ability to solve quadratic equations in terms of sine and find all solutions in a given interval.

Key Terms and Formulas

• Quadratic equations: $ax^2 + bx + c = 0$

• Factoring or quadratic formula

• Unit circle values for sine

Step-by-Step Guidance

1. Let $y = \sin x$ and rewrite the equation as a quadratic in $y$.

2. Factor or use the quadratic formula to solve for $y$.

3. For each solution for $y$, solve $\sin x = y$ for $x$ in $[0, 2\pi)$.

Try solving on your own before revealing the answer!

Q13. Solve on $[0, 2\pi)$: $\tan(3x) = 1$

Background

Topic: Solving Trigonometric Equations with Multiple Angles

This question tests your ability to solve equations involving tangent and multiple angles, and to find all solutions in a given interval.

Key Terms and Formulas

• General solution for $\tan \theta = 1$: $\theta = \frac{\pi}{4} + k\pi$

• Solving for $x$ when $3x = \theta$

Step-by-Step Guidance

1. Set $3x = \frac{\pi}{4} + k\pi$ for integer $k$.

2. Solve for $x$ by dividing both sides by 3.

3. List all solutions for $x$ in $[0, 2\pi)$ by plugging in appropriate values for $k$.

Try solving on your own before revealing the answer!

Q14. Solve: $2 \sin^2 x - 1 = 0$

Background

Topic: Solving Quadratic Trigonometric Equations

This question tests your ability to solve for $x$ in a quadratic equation involving sine.

Key Terms and Formulas

• Quadratic equations

• Unit circle values for sine

Step-by-Step Guidance

1. Isolate $\sin^2 x$ by adding 1 to both sides and dividing by 2.

2. Take the square root of both sides to solve for $\sin x$.

3. Find all $x$ in $[0, 2\pi)$ such that $\sin x$ equals the resulting values.

Try solving on your own before revealing the answer!